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In supergravity, there are 3-form fields known as C fields.

However, according to string theory, the fundamental degrees of freedoms are from string worldsheets (so at most they are fundamentally 2d).

My question is that how can the fundamental 2d strings construct the 3d or 3-form fields known as $C$ fields living on 3-manifolds (or homology group of 3-manifolds)?

  • otherwise, do we need to introduce 3d branes that are not from 2d string worldsheets?
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    $\begingroup$ 11-dimensional supergravity is the low energy effective limit of M-theory. M-theory has no string content - instead, the fundamental objects are the M2-brane, M5-brane. The three-form C-field then couples electrically to the worldvolume of the M2-brane and magnetically to the worldvolume of the M5-brane. $\endgroup$ Jun 21 at 2:22
  • $\begingroup$ How about from 12-dimensional (supergravity?) as the low energy effective limit of F-theory?? Does F theory have strings or higher branes? $\endgroup$ Jun 21 at 2:57
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    $\begingroup$ I know next to nothing about F-theory, but I think it has the same content as type IIB string theory, so fundamental strings and odd-numbered branes. If you are still interested in the M-theoretic three-form field, the relevant object to look at in F-theory is the $G_4$ flux (there is no organic RR 3-form gauge field since there are no 2-branes). As for 10+2D SUGRA, I have no idea how useful that is. $\endgroup$ Jun 21 at 5:58
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I'm just going to consider the bosonic sector in the following. I think your issue comes from this:

the fundamental degrees of freedoms are from string worldsheets (so at most they are fundamentally 2d).

Recall that the sigma model $X^i :\Sigma \hookrightarrow M$ describes the embeding of strings into a spacetime $M$. The degrees of freedom are the fields $X^i$ themselves and not the coordinates on the worldsheet that are mapped to the target space.

The massless field content of type IIB string theory is described by type IIB supergravity in ten dimensions, whose bosonic field content in the Neveu-Schwarz– Neveu-Schwarz (NS-NS) and Ramond–Ramond (R-R) sectors. These fields are a backreation on the target space $M$ of the first excitation states and are described by an action on the target and not the worldsheet.

For example, the R-R p-forms $C_p \in \Omega^p(M)$. For a complete dictionary of the field content, see the democratic formulation.

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  • $\begingroup$ Isn't the question talking about the 11D SUGRA/M-theory 3-form field? $\endgroup$ Jun 23 at 3:07
  • $\begingroup$ Tbh I'm not sure what the question is, so I've answered the potential confusion that I quoted. And also I'm not super familiar with the M2/5 branes of M-theory. I can add that to my very long to-do list :p $\endgroup$ Jun 23 at 12:03

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